| L(s) = 1 | + (−8.71 + 5.60i)2-s + (3.84 − 26.7i)3-s + (−8.57 + 18.7i)4-s + (279. − 82.0i)5-s + (116. + 254. i)6-s + (998. + 1.15e3i)7-s + (−219. − 1.52e3i)8-s + (−699. − 205. i)9-s + (−1.97e3 + 2.27e3i)10-s + (−1.81e3 − 1.16e3i)11-s + (468. + 301. i)12-s + (2.12e3 − 2.45e3i)13-s + (−1.51e4 − 4.44e3i)14-s + (−1.11e3 − 7.78e3i)15-s + (8.71e3 + 1.00e4i)16-s + (−1.40e3 − 3.08e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.770 + 0.495i)2-s + (0.0821 − 0.571i)3-s + (−0.0670 + 0.146i)4-s + (0.999 − 0.293i)5-s + (0.219 + 0.480i)6-s + (1.09 + 1.26i)7-s + (−0.151 − 1.05i)8-s + (−0.319 − 0.0939i)9-s + (−0.624 + 0.720i)10-s + (−0.412 − 0.264i)11-s + (0.0783 + 0.0503i)12-s + (0.268 − 0.309i)13-s + (−1.47 − 0.433i)14-s + (−0.0855 − 0.595i)15-s + (0.532 + 0.614i)16-s + (−0.0695 − 0.152i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.54271 + 0.531920i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.54271 + 0.531920i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-3.84 + 26.7i)T \) |
| 23 | \( 1 + (-5.33e4 + 2.37e4i)T \) |
| good | 2 | \( 1 + (8.71 - 5.60i)T + (53.1 - 116. i)T^{2} \) |
| 5 | \( 1 + (-279. + 82.0i)T + (6.57e4 - 4.22e4i)T^{2} \) |
| 7 | \( 1 + (-998. - 1.15e3i)T + (-1.17e5 + 8.15e5i)T^{2} \) |
| 11 | \( 1 + (1.81e3 + 1.16e3i)T + (8.09e6 + 1.77e7i)T^{2} \) |
| 13 | \( 1 + (-2.12e3 + 2.45e3i)T + (-8.93e6 - 6.21e7i)T^{2} \) |
| 17 | \( 1 + (1.40e3 + 3.08e3i)T + (-2.68e8 + 3.10e8i)T^{2} \) |
| 19 | \( 1 + (-2.40e3 + 5.26e3i)T + (-5.85e8 - 6.75e8i)T^{2} \) |
| 29 | \( 1 + (-5.19e4 - 1.13e5i)T + (-1.12e10 + 1.30e10i)T^{2} \) |
| 31 | \( 1 + (2.14e3 + 1.49e4i)T + (-2.63e10 + 7.75e9i)T^{2} \) |
| 37 | \( 1 + (-5.12e5 - 1.50e5i)T + (7.98e10 + 5.13e10i)T^{2} \) |
| 41 | \( 1 + (-1.28e5 + 3.78e4i)T + (1.63e11 - 1.05e11i)T^{2} \) |
| 43 | \( 1 + (1.18e5 - 8.26e5i)T + (-2.60e11 - 7.65e10i)T^{2} \) |
| 47 | \( 1 - 1.21e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + (2.91e5 + 3.36e5i)T + (-1.67e11 + 1.16e12i)T^{2} \) |
| 59 | \( 1 + (-1.06e6 + 1.22e6i)T + (-3.54e11 - 2.46e12i)T^{2} \) |
| 61 | \( 1 + (3.00e5 + 2.09e6i)T + (-3.01e12 + 8.85e11i)T^{2} \) |
| 67 | \( 1 + (-2.58e6 + 1.65e6i)T + (2.51e12 - 5.51e12i)T^{2} \) |
| 71 | \( 1 + (3.07e6 - 1.97e6i)T + (3.77e12 - 8.27e12i)T^{2} \) |
| 73 | \( 1 + (1.03e5 - 2.27e5i)T + (-7.23e12 - 8.34e12i)T^{2} \) |
| 79 | \( 1 + (1.04e6 - 1.20e6i)T + (-2.73e12 - 1.90e13i)T^{2} \) |
| 83 | \( 1 + (-6.79e6 - 1.99e6i)T + (2.28e13 + 1.46e13i)T^{2} \) |
| 89 | \( 1 + (7.51e5 - 5.22e6i)T + (-4.24e13 - 1.24e13i)T^{2} \) |
| 97 | \( 1 + (1.28e7 - 3.75e6i)T + (6.79e13 - 4.36e13i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.28204694558601177270172441106, −12.48820406530274930816478319202, −11.14433900388082731774991242204, −9.496751334715871939085308031723, −8.655660280328388884963947760793, −7.83983595064797212315382540904, −6.29842248107279305102047696370, −5.13150895126797694973333128980, −2.60305777607899132855578205460, −1.11392627957320134903265461904,
0.961380325648667537061457654788, 2.25013672181355963897050489507, 4.38395276972956058072409045494, 5.68667513023624926098735071073, 7.54622485790994482928792245656, 8.866474604082116868266716908138, 10.01264415875788213836753290647, 10.58712156235527484974527300164, 11.45432967447159016958407670229, 13.52711903498885854193623357445
